Moduli of smoothness and growth of the Laguerre transform: two-sided estimates

Abstract

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge 1$ and $1\le p\le 2$, a weighted ${\ell }^{p^{\prime }}$ norm of $\min \{1,nt{\}}^r{\hat{f}}_{\alpha }(n)$ is bounded by the modulus of order $r$; for $2\le p\le \infty$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff--Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<\beta <r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.

arXiv comment31 pages